Individual vs. Aggregate Preferences:
The Case of a Small Fish in a Big Pond
Douglas W. Blackburn∗
Indiana University
Andrey D. Ukhov†
Indiana University
October 2006
Abstract
We show that there can be significant difference between risk preferences of individual investors and aggregate preferences toward risk in the economy. To demonstrate this, we investigate aggregate properties of an economy where all investors have convex utility functions
corresponding to risk seeking behavior. In the case of risk seeking individual agents with identical
initial endowments, assuming a budget constraint, and facing perfect competition the aggregate
economy is risk neutral. In the case of risk seeking individuals with different initial endowments,
we show that if there exists a continuum of wealth classes the economy in the aggregate will
exhibit risk averse behavior. Thus, an economy consisting of risk seeking agents can lead to
an aggregate economy that is risk averse. We prove that the converse is also true. For an
economy that in the aggregate exhibits risk aversion we can construct an economy of all risk
seeking agents that in the aggregate produces the given risk averse indifference curve. That is,
an economy demanding a risk premium can be formed from individuals who do not demand
such compensation. In the context of the equity premium puzzle our analysis shows that for
an aggregate economy to display a high degree of risk aversion and to demand a relatively high
risk premium individuals do not need to have implausibly high risk aversion.
JEL Classification: G10, D11, D80.
Keywords: Risk aversion, risk seeking, investor sentiment, risk premium.
∗
Douglas W. Blackburn, Kelley School of Business, Indiana University, 1309 East 10th Street, Bloomington, IN
47405. E-mail: [email protected] We would like to thank Peter Bossaerts, Ned Elton, Michael Gallmeyer, John
Geanakoplos, Craig Holden, Jonathan E. Ingersoll, Jr., Robert Jennings, Stavros Panageas (discussant), Thierry
Post, Jun Qian, Robert Savickas, Matthew Spiegel, Charles Trzcinka, Gregory Udell, participants of the Frontiers of
Finance 2005, 2006 FMA Meeting (Salt Lake City), and the Washington Area Finance Association 2005 conferences
for helpful comments. This paper is a revised version of the working paper previously distributed under the title
“Equilibrium Risk Premia for Risk Seekers.”
†
Corresponding author: Andrey D. Ukhov, Kelley School of Business, Indiana University, 1309 East 10th Street,
Bloomington, IN 47405. Telephone: 812-855-2698, Fax: 812-855-5875. E-mail: [email protected]
Individual vs. Aggregate Preferences:
The Case of a Small Fish in a Big Pond
Abstract
We show that there can be significant difference between risk preferences of individual investors
and aggregate preferences toward risk in the economy. To demonstrate this, we investigate aggregate
properties of an economy where all investors have convex utility functions corresponding to risk
seeking behavior. In the case of risk seeking individual agents with identical initial endowments,
assuming a budget constraint, and facing perfect competition the aggregate economy is risk neutral.
In the case of risk seeking individuals with different initial endowments, we show that if there exists
a continuum of wealth classes the economy in the aggregate will exhibit risk averse behavior. Thus,
an economy consisting of risk seeking agents can lead to an aggregate economy that is risk averse.
We prove that the converse is also true. For an economy that in the aggregate exhibits risk aversion
we can construct an economy of all risk seeking agents that in the aggregate produces the given
risk averse indifference curve. That is, an economy demanding a risk premium can be formed from
individuals who do not demand such compensation. In the context of the equity premium puzzle
our analysis shows that for an aggregate economy to display a high degree of risk aversion and
to demand a relatively high risk premium individuals do not need to have implausibly high risk
aversion.
JEL Classification: G10, D11, D80.
Keywords: Risk aversion, risk seeking, investor sentiment, risk premium.
1
Introduction
Many asset pricing models are based on the idea that a representative agent prices all assets
in an economy. This representative agent is the one price setter and is the one whose optimal
portfolio is the aggregate market. Theory has tended to model the representative agent using simple,
differentiable utility functions such as the power utility or the exponential utility. Models using
these simple utility functions, however, have not been able to describe such statistical characteristics
as the means and variances of asset returns and this has lead to a large literature on puzzles and
anomalies.1
In response to these puzzles, many researchers have sought to better understand how individuals
make decisions in hopes of explaining the pricing puzzles. The psychology literature has revealed
that individuals are averse to losses, tend to be overconfident, under- or overreact to different news
events, choose consumption based on habit, and exhibit other behavioral patterns. Kahneman and
Tversky (1979) observed that individuals underweight uncertain outcomes and overweight certain
outcomes. As a result, individuals are risk averse over certain gains and risk seeking over certain
losses. Friedman and Savage (1948), and subsequently Markowitz (1952), observed that individuals
buy both lottery tickets and insurance. They argue that agents must be risk seeking over a range
of gains and risk averse over a range of losses.2
These individual behavioral biases along with other observations concerning individuals have
leaped into the asset pricing research. Jarrow (1988) shows that the Arbitrage Pricing Theory
holds even when agents do not have preferences that can be represented by risk-averse expected
utility functions. Jarrow and Zhao (2006) study optimal portfolios of investors with downside
1
2
See Constantinides (2002) and Hirshleifer (2001) for a review.
See Brunk (1981), Gregory (1980), Hakansson (1970), and Kwang (1965) for studies of Friedman and Savage
utility function.
1
loss-averse preferences. Shefrin and Statman (2000) develop a positive behavioral portfolio theory
(BPT) where investors choose portfolios by considering expected wealth, desire for security and
potential, aspiration levels, and probabilities of achieving aspiration levels. The optimal portfolios
of BPT investors resemble combinations of bonds and lottery tickets, consistent with Friedman
and Savage’s observation. Bakshi and Chen (1996) study, both theoretically and empirically, an
economy where investors acquire wealth not only for its implied consumption, but for the resulting
social status. When investors care about relative social status, propensity to consume and risk
taking behavior depend on social standards, and this has an impact on asset prices. Levy and Levy
(2002) find strong empirical support for Markowitz utility function in their experimental study.
Post and Levy (2005), using stochastic dominance criteria, find that the Markowitz utility function
(risk aversion of losses and risk seeking over gains) captures the cross-sectional pattern of stock
returns. Constantinides (1990) shows how habit formation helps resolve the equity premium puzzle.
Goetzmann and Massa (2002) identify a group of index fund investors who systematically invest
in the fund after volatility increases. Coval and Shumway (2005) find evidence that proprietary
traders are loss averse (risk averse over gains and risk seeking over losses). Many others have found
evidence of risk seeking behavior in aggregate prices (Jackwerth 2000; Levy and Levy 2001, 2002;
Rosenberg and Engle 2002; Kliger and Levy 2002; Bakshi and Wu 2006; Green and Rydqvist 1997;
Florentsen and Rydqvist 2002) and the evidence spans time, several countries and many different
empirical methods.
Two challenges arise. The first concerns the failure of classical models to explain the equity
premium. Mehra and Prescott (1988) find that the risk averse agent must have a very large risk
aversion coefficient in order to explain the large equity premium observed in the data. The agent
must be very risk averse and demand a large compensation for risk. What if individual agents
actually do have a Kahneman-Tversky or a Markowitz utility function and are risk seeking over
some ranges of wealth? If this is the case, then the equity premium puzzle may be a much larger
problem than originally thought.
The second challenge is in regard to the connection between individual behavior and aggregate
behavior. The behavioral finance literature describes the behavioral patterns of individuals and
how their financial decisions are affected by their behavior. Kahneman and Tversky’s prospect
theory is about individual behavior. Friedman and Savage (1948) argue that individuals gamble
2
and buy insurance. Individuals may be loss averse, overconfident, and form consumption habits.
What is unclear is how individual behavior affects aggregate prices. On the other hand, many
have found evidence of unusual behavior in aggregate prices. For example, Post and Levy (2005)
find the Markowitz utility function best explains the cross section of returns. Nonetheless, can we
necessarily imply anything about individual behavior from these findings? By observing behavior
in aggregate prices, can we conclude that individuals have the same behavior?
We show in this paper that the representative agent of the economy can be very different from
the individuals who make up the economy. First, individual behavior does not necessarily lead
to the same aggregate behavior. Second, aggregate behavior does not necessarily imply anything
concerning the behavior of individuals. Further, individual risk seeking behavior does not necessarily challenge a positive equity premium. We consider an economy comprised solely of risk seeking
agents who have utility functions defined over current consumption and consumption at a future
date. Surprisingly, the aggregate economy is risk averse. We do not believe or suggest that all individuals are actually risk seeking. We use this extreme assumption in order to contrast individual
and aggregate behavior.
Beginning with the case of risk seeking agents facing a budget constraint, we prove that although
individual agents are risk seekers, under perfect competition and identical initial endowments, the
aggregate economy is risk neutral. Next, we consider the case of risk seeking agents with different
initial endowments. If there exists a closed and bounded continuum of wealth classes then the
economy’s representative agent will exhibit risk averse behavior.
Finally, we prove that for any aggregate economy that demands a risk premium, there exists
a collection of risk seeking individuals with a specifically chosen wealth distribution that exactly
replicates the given risk averse economy. This finding is relevant for the studies that aim at using
the data on aggregate behavior (aggregate consumption, market returns, market volatility) to draw
conclusions about individual behavior. An example if such research question is whether we can
necessarily say that since we observe a large positive market risk premium that all (or most) agents
in the economy are highly averse to risk. The answer to this is no. To show this, we prove that
an economy exhibiting aggregate risk averse behavior can be supported by risk-seeking individuals.
Though it may be tempting to make statements concerning individual behavior based on aggregate
data, the analysis in this paper shows that the relationship between groups and individuals is not
3
clear or direct. An economy demanding a risk premium can be formed from individuals who do
not demand such compensation. Our analysis also implies that for an aggregate economy to display
a high degree of risk aversion and to demand a relatively high risk premium, individuals do not
need to have implausibly high risk aversion.
A simple example helps gain intuition. Consider a large economy of identical risk-seeking
individuals with identical endowments across two states. Agents’ preferences across the two states
are described by the time separable utility function E[U (C1 ) + U (C2 )] where C1 = C2 = C is
the aggregate consumption, U (·) is the agents’ risk-seeking utility function and E[·] denotes the
expectation operator. In this simple setting, prices of the two states are equal.
Therefore, half
the population will enter a bet that pays 1 in state one and zero in state two. The other half of
the population will take the opposite bet. Interestingly, though each individual is risk-seeking, the
economy behaves as if it is risk-neutral.
The remainder of the paper is organized as follows. Section 2 develops the aggregation results.
Subsection 2.2 studies the case of identical agents. Subsection 2.3 analyzes the case when agents
have different initial wealth. Section 3 shows that every risk averse (in the aggregate) economy can
be obtained by aggregating risk seeking individual agents. Section 4 concludes. All proofs are in
the Appendix.
2
The Model of Utility Aggregation
2.1
Convex Utility
In a typical model, an agent solves the expected utility maximization problem,
h
i
e1 ) ,
max E u(C0 , C
subject to a budget constraint, where the utility of consumption is assumed concave and additively
separable:
u(C0 , C1 ) = U0 (C0 ) + U1 (C1 ) .
In this paper we depart from the concavity (risk aversion) assumption made about the utility
function of individual agents in the economy. Several landmark papers argue that individual utility
4
functions have a convex region. We take this assumption to the extreme by studying aggregate
properties of an economy where all individuals have identical convex additively separable utility of
consumption:
u (xi , yi ) = U0 (xi ) + U1 (yi ) ,
where xi represents agent i’s consumption at time 0 and may be referred to as the “X consumption
good,” and yi represents agent i’s (uncertain) consumption at time 1 and may be referred to as the
“Y consumption good.” We assume that u (xi , yi ) is twice continuously differentiable and increasing
in all arguments. Without loss of generality, we work with indifference curves. Individual agents are
assumed to have convex utility functions, which correspond to concave indifference curves (Figure
1). We will first show that when all agents are identical that the aggregate economy is characterized
by linear indifference curves, which correspond to risk neutral preference. We will then show that
when all agents have the same utility but are heterogeneous in wealth that the aggregate economy
is characterized by convex indifference curves.
2.2
Identical Agents
We begin with an economy with N identical agents all with the same utility function u (xi , yi ) and
all having the same initial level of wealth. Each agent has a strictly risk seeking utility function
that corresponds to strictly concave indifference curves. The economy is initially endowed with a
fixed quantity of Y , Ymax > 0, where Ymax is the maximum amount of the Y good that the economy
can produce. None of the X consumption good is available in this initial case. Since all agents
are the same then it is clear that all agents will be initially endowed with an equal quantity of Y .
In addition, all agents will be equally happy since they have the same utility, u(xi , yi ) = k, brought
about by having the same initial allocation of the Y good.
Suppose, instead, there is a positive amount of the X consumption good, X > 0, available for
allocation. We again want to determine the efficient allocation of both the X and the Y goods.
By efficient we mean for a fixed quantity of the X good, the economy should produce the minimum
amount of Y necessary so that all agents have utility equal to k as in the previous case. Since the
X good is now available, less Y is needed in the economy. For each X > 0, we want to find the
minimum Y required so that all agents utility remains at k and all goods are completely allocated.
5
By finding all such (X, Y ) pairs we are essentially tracing the aggregate economy’s indifference
curve.3 To do this, we must solve the optimization problem:
Min
x1 ,...,xN
subject to
Y =
N
P
N
P
Ψ (xi ; k)
i=1
xi = X
(1)
i=1
and
xi ≥ 0, yi ≥ 0 for all i.
The capitalized Y and X is used to denote aggregate quantities of the two goods whereas
the lower case x and y are used to describe the quantity of the two goods held by individual
agents. By the Implicit Function Theorem, the equation u (xi , yi ) = k defines the implicit function,
yi = Ψ (xi ; k). We do assume no short selling in our problem. However, our results hold for any
finite short selling constraint since we can redefine the coordinate axis through a change of variable.
This is a reasonable constraint since no agent can realistically short an infinite amount even if he
does choose to “bet the farm”.What is important, is to rule out infinite borrowing, not borrowing
per se.
In order solve Problem 1, we must identify the quantity of X and Y allocated to each agent
that maintains each agents’ utility at k. In determining the optimal allocation we must consider
several possibilities.
The X good could be equally distributed among all agents or the X good
could be allocate unequally among all agents. It turns out that the second alternative is the most
efficient. To see why, consider Figure 1. Figure 1 shows an indifference curve for a risk seeking
agent.
A single agent can at most hold ymax of Y and xmax of X and he is indifferent between
holding ymax and holding xmax .
The agent is allocated less Y for each unit of X he receives so
as to stay on the kth indifference curve. But, the marginal rate of substitution increases with the
each unit of X received. Suppose he receives xA . The slope of the tangent line indicates the trade
off between X and Y . Notice that the slope is very small indicating that the next unit of X does
not greatly decrease his allocation of Y . However, suppose he receives xB . Now, the slope of the
tangent is much steeper indicating that each additional unit of X greatly decreases his allocation of
Y . What we learn from this picture is that the economy is better off allocating a large quantity of
3
As previously stated, the term “aggregate indifference curve” refers to the indifference curve for the aggregate
economy, which gives the rate at which the total amount of consumption in one period must increase when the
aggregate amount of consumption in the other period falls, in order to keep utility levels of all agents unchanged.
6
X to one individual since the trade off between the goods is high. The trade off is at its maximum
when the agent is allocated xmax . So, the economy is required to produce the least amount of Y
when all the X good is allocated to only a few agents so that these agents hold none of the Y good.
One agent will likely hold a remainder of X in the case that there is not enough X for the agent
to receive the entire xmax . The remaining agents will hold all Y and no X. All agents will either
specialize in the X or the Y good but one agent will be forced to hold some of both (the remainder
of X and enough Y to yield a utility of k). This leads us to the formal statement of Proposition 1.
Proposition 1 (Efficient Allocation) Let X = (n − 1)xmax + r, for n = 1, 2, ..., N and r ∈
(0, xmax ] be the quantity of X available and define xmax as such value of x that solves: u (x, 0) = k.
Then xmax is the maximum amount of X each individual agent can hold and ymax = Ψ (0; k) is the
maximum amount of Y the agent can hold. The solution to (1) is
⎧
⎪
⎪
(x , 0)
for i = 1 to n − 1
⎪
⎨ max
(xi , yi ) =
(r, Ψ (r; k)) for i = n
⎪
⎪
⎪
⎩
for i = n + 1 to N.
(0, ymax )
As previously mentioned, the no short sale constraint can be exchanged with a finite short sale
constraint. In this case we would see a similar allocation. Some agents would short the maximum
amount permitted of the X good and hold more than 100% of Y . The remaining agents would
short the maximum quantity of Y and hold more than 100% of X. One agent would be allocated
some of both goods.
Now that the efficient allocation for each agent has been determined, we can turn our attention
to aggregating the demand under perfect competition. This is found by aggregating the demands
from Proposition 1. For a given quantity of X, we write the total quantity of Y required by the
economy as
YN (X)
=
i´
³
h
X
ymax , for X ∈ [0, Xmax ] .
Ψ (r; k) + N − 1 − int xmax
There are two types of agents who hold Y . There are N − n − 1 agents who specialize in Y , and
there is one agent who holds some of X and Y (the nth agent). This equation sums together the
quantity of Y required by the nth agent and the quantity of Y required by those holding ymax (ymax
times the number of agents).
7
To determine the demand under perfect competition, we consider the limit of the aggregate
demand function as the number of agents, N, increase to infinity while the supply of Y and X is held
constant. By taking the limit in this fashion we keep the focus on the increasing competitiveness
of the economy. Perfect competition is modeled by increasing the number of agents in such a
way that the influence of each individual participant becomes negligible. This notion of perfect
competition is consistent with Aumann (1964) who formally argues that a mathematical model of
perfect competition should contain infinitely many participants.
The previous proposition establishes that all but one agent, agent n, will be endowed with
either all Y or all X.
Hence, all but one agent specialize in either X or Y .
As the number of
agents increase to infinity, agent n’s contribution to the economy becomes negligible. Therefore,
in perfect competition, all agents are at a corner with the exception of the one negligible agent a set of measure zero. The indifference curve that describes this economy’s behavior is a straight
line connecting the two points (Xmax , 0) and (0, Ymax ).
Figures 2 and 3 illustrate these arguments.
for the economy with two agents.
X.
Figure 2 shows the aggregate indifference curve
Suppose XA of X is available.
This is a small quantity of
From Proposition 1, we know that it is most efficient to allocate X to one agent until that
agent holds xmax . But there is not enough X for agent 1 to specialize in X. Therefore, agent 1
is allocated all of XA and enough Y to keep him on the k th indifference curve, while agent 2 holds
all Y . This yields the aggregate indifference point indicated by point A. In this case, the shape
of the aggregate indifference curve depends only on the first agent since he is the only one with an
interior position. This holds true until enough X is available for agent 1 to specialize in X, XB .
Now, agent 1 holds all X and agent 2 holds all Y . If more X is available, then the curvature of
agent 2’s indifference curve will define the shape of the aggregate indifference curve. Agent 1 will
no longer matter since he does not hold an interior position.
Figure 3 shows the case of five agents. Recall that Xmax and Ymax are held constant. Therefore,
it takes less X for an agent to specialize in X. We now see five “lumps” representing the five agents.
It is still the case that the curvature of the aggregate indifference curve is controlled by the one
agent holding the interior position. But with five agents, the “lumpiness” is much smaller than
in the two agent case. As the number of agents increase, the lumpy aggregate indifference curve
will converge to the straight dotter line. That is, the contribution of the one individual with the
8
interior position becomes less significant in comparison to the many agents specializing in either X
or Y .
Proposition 2 formally establishes this claim.
The details of the proof are provided in the
appendix.
∗
max
Proposition 2 For fixed integer N ∗ , as z → ∞, Yz converges to Y∞ (X) = − X
Ymax X + N Ψ (0; k)
uniformly on [0, Xmax ] where Xmax = N ∗ Ψ−1 (0; k), and Ymax = N ∗ Ψ (0; k).
In Proposition 2, we start with an economy with a fixed number of agents, N ∗ , and this economy
achieves some initial level of individual utility. Then we show that as the economy approaches
perfect competition, the aggregate indifference curves become straight lines. This proposition states
that although individual agents have concave indifference curves, the aggregate economy, under
perfect competition and identical initial wealth endowments, has a convex (linear) indifference
curve. Though individual agents are risk-seeking, the aggregate economy behaves as if it is riskneutral.
2.3
Agents with Different Initial Wealth
We now consider the case where the agents all have the same convex utility function, but initial
wealth is different across agents. Each agent will be initially endowed with a different quantity of
Y.
We begin with a simple case. Suppose there are N1 + N2 agents divided into two wealth classes.
There are N1 type 1 agents (the poor) who have initial wealth less than the N2 type 2 agents (the
rich). We first consider the case where the economy produces Ymax and no X is available. Wealth
is divided in such a way that the efficient allocation of Ymax between the two types causes the type
1 agent to be on the indifference curve k1 and the type 2 agent to be on the k2 indifference curve
where k1 < k2 . The Y good is initially allocated efficiently. Hence, all poor agents are allocated
the same quantity and all the rich agents are allocated the same quantity but the rich are given
more than the poor agents. We now ask the same question posed previously. If X ∈ (0, Xmax ] is
available for allocation, what is the minimum amount of the Y good required in the economy in
order for the poor to always be on the k1th indifference curve and for the rich to remain on the
k2th indifference curve? We are searching for all the efficient (X, Y ) allocations that maintain all
9
agents’ level of utility.
This question is exactly the same as the previous problem except that agents are no longer
identical.
As before, due to the concavity of the indifference curves, the efficient allocation will
be a corner solution. In the previous problem, we allocated all the X good to a set of agents so
that all agents held all X or all Y and only one agent held a combination of both. This procedure
resulted in a correct allocation because the rate at which each agent was willing to substitute Y
for X increased with X.
Each agent was willing to pay more, in units of the Y good, for each
additional increment of the X good. The same is true in this case except we must be careful in
selecting which type, the rich or the poor, should be allocate the X first. That is, which type, the
rich or the poor, has the greatest rate of substitution for each increment of X? The answer to this
¯
¯
¯
¯
¯ y2,max ¯
¯
¯y
>
question depends on the concavity of the indifference curve. If ¯ x1,max
¯
¯
x2,max ¯ then the type
1,max
1 agent’s (the poor) rate of substitution is greater and they should be given the X good first (see
¯
¯ ¯
¯
¯y
¯ ¯ y2,max ¯
<
Figure 4). If ¯ x1,max
¯ ¯ x2,max ¯ then the rich are willing to pay more for the X good and should be
1,max
¯ ¯
¯
¯
¯ y1,max ¯ ¯ y2,max ¯
given X first . If, however, ¯ x1,max ¯ = ¯ x2,max ¯ then it does not matter which type receives the X
good first and the problem reduces to the previous identical agent case. For the remainder of this
¯
¯
¯
¯
¯ y2,max ¯
¯y
¯
>
section we consider the case ¯ x1,max
¯
¯
x2,max ¯. An example of a simple convex utility function
1,max
that satisfies this condition is u (xi , yi ) = αx2i + βyi4 , α > 0, β > 0. This is, of course, not the only
utility function that satisfies this condition.
case
y1,max
x1,max
<
Though not discussed here, all results hold for the
y2,max
x2,max .
Without loss of generality, we order the agents such that i = 1, ..., N1 are type 1 and i = N1 +
1, ..., N1 + N2 are type 2. All agents have identical twice continuously differentiable, monotonically
increasing convex utility function. Type 1 agents achieve utility level k1 and type 2 agents achieve
utility level k2 . By the Implicit Function Theorem, the equation u (xi , yi ) = k defines the implicit
function, yi = Ψ (xi ; k). The problem to be solved is
Min
x1 ,...,xN1 +N2
subject to
Y =
N1
P
Ψ (xi ; k1 ) +
i=1
N1P
+N2
N1P
+N2
Ψ (xi ; k2 )
i=N1 +1
xi = X
(2)
i=1
and
xi ≥ 0, yi ≥ 0 for all i.
This problem is similar but slightly more difficult than the previous problem. However, we can
apply the intuition learned from the previous case. As before, for a fixed quantity of X, we want to
10
produce the minimum amount of Y necessary to keep the type 1 agents on the k1 indifference curve
and the type 2 agents on the k2 indifference curve. Because the marginal rate of substitution is
greatest for the poor agents (by assumption), allocating the X good to the poor first will decrease
the amount of Y needed in the economy more than if the X good was first allocated to the rich.
This leads us to two possibilities: if X is less than what is needed for all the poor agents and if
X is more than what is needed for all the poor agents. In the first case, the rich will not receive
any of the X good since the X good is first allocated to the poor. Instead, the poor will receive
all the X good in the same way as described in Proposition 1. In the second case, all the poor
agents specialize in the X good.
The remaining X will be allocated to the rich as described in
Proposition 1. This allocation is formalized in Proposition 3.
Proposition 3 Let the number of type 1 and type 2 agents be N1 and N2 , respectively.
The
type 1 agents have an initial utility of k1 and the type 2 agents have initial utility k2 > k1 and
¯
¯
¯
¯
¯
¯ y2,max ¯
¯y
−1 (0; k ) and y
>
If
assume that ¯ x1,max
¯
¯
i
i, max = Ψ (0; ki ) for i=1,2.
x2,max ¯ where xi, max = Ψ
1,max
X = (n − 1)x1,max + r ≤ N1 x1, max , where n = 1, 2, ..., N1 , and r1 ∈ (0, x1,max ], then the solution to
problem (2) is
(x1,i , y1,i ) =
⎧
⎪
⎪
(x
, 0)
⎪
⎨ 1,max
(r1 , Ψ (r1 ; k1 ))
⎪
⎪
⎪
⎩ (0, y
1,max )
(x2,i , y2,i ) = (0, y2,max )
for
for
i = 1 to n − 1
for
i=n
for
i = n + 1 to N1
i = N1 + 1 to N1 + N2 .
If N1 x1, max > X ≥ N1 x1, max + N2 x2, max and X = N1 x1, max + (n − 1)x2,max + r2 where r2 ∈
(0, x2,max ] and then the efficient allocation is
(x1,i , y1,i ) = (x1,max , 0) for i = 1, ..., N1
⎧
⎪
⎪
(x
, 0)
for i = N1 + 1 to N1 + n − 1
⎪
⎨ 2,max
(x2,i , y2,i ) =
(r2 , Ψ (r2 ; k2 )) for i = N1 + n
⎪
⎪
⎪
⎩ (0, y
for i = N1 + n + 1 to N1 + N2
2,max )
Notation xi, max = Ψ−1 (0; ki ) means that xi, max is the value of x that solves u (x, 0) = ki for i=1,2.
The efficient allocation in Proposition 3 is very similar to the allocation described in Proposition 1.
All but one agent holds either all X or all Y .
11
It should not be surprising, then,
that the aggregate indifference curves under perfect competition are very similar to the those described in Proposition 2.
Whereas Proposition 2 describes a linear aggregate indifference curve
Ymax
, the aggregate indifference curve here is piecewise linear where the first linwith slope − X
max
y
y
for X ∈ [0, N1 x1, max ) and the second piece has slope − x2,max
for
ear segment has slope − x1,max
1,max
2,max
¯
¯
¯
¯
¯ y2,max ¯
¯ y1,max ¯
X ∈ (N1 x1, max , N1 x1, max + N2 x2, max ]. Since, by assumption, ¯ x1,max ¯ > ¯ x2,max ¯, the two linear
segments join to form convex aggregate indifference curves.
The aggregate indifference curves,
therefore, are continuous and convex but not differentiable at the point where the two linear segments connect (see Figure 5). Such an aggregate indifference curve represents an economy that is
risk averse! We now formalize this.
Proposition 4 The aggregate demand of an economy with N1 type 1 agents with utility k1 and N2
type 2 agents with utility k2 such that k1 < k2 is
⎧
⎨ − Ψ(0;k1 ) N1 Ψ−1 (0; k1 ) + N2 Ψ (0; k2 ) ,
Ψ−1 (0;k1 )
Y (X) =
⎩ − Ψ(0;k2 ) ¡X − N Ψ−1 (0; k )¢ + N Ψ (0; k ) ,
1
1
2
2
Ψ−1 (0;k2 )
X ∈ [0, N1 x1, max ]
X ∈ [N1 x1, max , N1 x1, max + N2 x2, max ]
where N1 x1, max = N1 Ψ−1 (0; k1 ) and N2 x2, max = N2 Ψ−1 (0; k2 ).
We now extend this result to n wealth classes. For simplicity, we assume that all wealth classes
have the same number of agents, Ni = N for all i. This is without loss of generality. The agents
are ordered by indifference curve, ki < ki+1 . Then, for wealth classes i = 1 to n, the system of
linear equations is:
Yi (X) = − xyi,max [X − xi−1 ] + yi−1
i,max
where xi = N
i
P
j=1
xj,max and yi = Ymax − N
i
P
for
(3)
X ∈ [xi−1 , xi ] ,
yj,max with x0 = 0 and y0 = Ymax . The maximum
j=1
amount of X and Y that this economy can hold is Xmax = N
n
P
j=1
xj,max and Ymax = N
n
P
yj,max .
j=1
This system of equations creates a family of continuous, convex, but not differentiable indifference
curves.
The indifference curves are not differentiable at the points where the linear segments
connect. However, the indifference curves can be made smooth by infinitely increasing the number
of wealth classes so as to create a continuum of k values over a range kmin to kmax while holding Xmax
and Ymax constant. That is, for every indifference curve k ∈ [kmin , kmax ], there exists agents with
12
the appropriate level of wealth that places them on that indifference curve. Adding wealth classes
in this way reduces the length of each line segment. If there are an infinite number of wealth classes
creating, then the aggregate indifference curve will become smooth.
Whereas Figure 5 shows a
piecewise linear aggregate indifference curve, Figure 6 shows a much smoother curve that is created
through the addition of wealth classes. The following Theorem proves that the resulting aggregate
indifference curve will be convex and differentiable. Hence, such an economy will demand a risk
premium.
Theorem 5 Let there exist [kmin , kmax ], a continuum of utility levels with kmin > 0 and kmax < ∞,
and for each level of utility ki ∈ [kmin , kmax ] there exists an infinite number of agents so that the i−th
¯
¯ ¯
¯
¯y
¯ ¯ yi+1,max ¯
>
agent type has a demand function described by (3). If, for all ki < ki+1 , ¯ xi,max
¯
¯
xi+1,max ¯ and
i,max
fixing the maximum amount of X and Y at Xmax and Ymax , respectively, the aggregate indifference
curve for the economy is strictly convex and differentiable over [0, Xmax ]. If for all ki < ki+1 ,
¯ ¯
¯
¯
¯ yi,max ¯ ¯ yi+1,max ¯
=
¯ xi,max ¯ ¯ xi+1,max ¯then the aggregate indifference curve is convex (linear).
With two wealth classes, the economy behaves as if it has a piecewise linear, convex indifference
curve.
This indifference curve has two pieces and is not differentiable at the point where the
two linear segments connect.
By increasing the number of wealth classes, the number of linear
segments increase and the length of each segment decreases.
This theorem states that if there
exists a closed and bounded continuum of wealth classes then the aggregate indifference curve is
strictly convex and differentiable. Any gaps in wealth will result in a non-differentiable kink in the
aggregate indifference curve, but the curve will still be convex.
There is one issue that requires a comment. We have shown that the aggregate economy’s utility
function is quasiconcave but this does not necessarily lead to a concave utility function. Assuming
that the aggregate economy’s utility is also time additive (as we assumed with individuals), if
u(C0 , C1 ) = U0 (C0 ) + U1 (C1 )
is quasiconcave then at least either U0 or U1 must be concave (see Yaari 1977 for proof). If we
assume that U0 and U1 are from the same family of utility functions (both are power utilities),
then it is necessarily true that both must be concave and hence u(C0 , C1 ) is concave. However, if
we assume that U0 and U1 are independent then one of them might not be concave. In this case,
13
as Gorman (1970) argues, there will be“considerable regions” over which one of the consumption
goods will be inferior. In our economy, however, we are giving the economy the choice between
consuming today and consuming tomorrow. To consider one of the two consumption goods to be
inferior over large regions does not make sense in this setting. Hence, it is reasonable to assume that
both U0 and U1 are concave causing u(C0 , C1 ) to be concave. Therefore, though each individual
agent is risk-seeking, the economy behaves as if it is risk-averse! Preferences toward risk are very
different at the individual and at the aggregate levels.
3
De-Aggregating the Economy
We have shown that an economy consisting of risk seeking agents can lead to an aggregate economy
that is risk averse. This, of itself, is an interesting result since it calls into question how researchers
apply individual behavior in an economy where aggregate behavior is observed. But, we can still
take this analysis one step further. Since we observe aggregate behavior (aggregate consumption,
aggregate prices, etc.), can we necessarily use this data to make statements regarding individual
behavior? Can we necessarily say that since we observe a positive risk premium that all (or most)
agents in the economy are risk averse? The answer to this is no.
To show this, we simply need to perform the above analysis backwards.
We start with an
aggregate convex indifference curve, implying that the economy is risk averse.
We prove the
existence of a risk seeking agents that aggregate to the same risk averse aggregate indifference curve.
For any decreasing, convex function g : [0, Xmax ] → [0, Ymax ], there exists a distribution of wealth
classes that replicates the convex curve. Said differently, any economy exhibiting aggregate riskaverse behavior can be supported by risk-seeking individuals. All that is needed is an appropriate
wealth distribution.
Theorem 6 Consider an economy of risk averse agents with aggregate demand defined by Y =
g(X) with the properties g 0 (X) < 0, g 00 (X) > 0, 0 < g 0 (X) < ∞ for all X ∈ [0, Xmax ], g(0) = Ymax
and g(Xmax ) = 0. There exists a distribution of wealth such that an economy of risk seeking agents,
¯
¯
¯
¯
¯
¯ yi+1,max ¯
¯y
>
with indifference curves Y = Ψ (X; ki ) satisfying the property ¯ xi,max
¯
¯
xi+1,max ¯ for ki < ki+1 ,
i,max
has the same aggregate demand as the economy of risk averse agents.
Though it may be tempting to make statements concerning individual behavior based on ag14
gregate data, this analysis shows that the relationship between groups and individuals is not clear
or direct.
An economy demanding a risk premium can be formed from individuals who do not
demand such compensation.
Our results are staggering when contrasted with a more conventional intuition. Suppose we assume that the representative agent of the economy has, for example, a negative exponential utility
function. By observing returns on the market we infer that the risk aversion of the representative
investor must be some value, RAA . According to arguments that lead to the equity premium puzzle, judgment may be applied to this value leading us to a conclusion that the level of risk aversion
of the representative agent is too high. This puzzle is further deepened if we assume that the individuals that make up the economy also have negative exponential utility. When all individuals have
negative exponential utility, the representative investor also has negative exponential utility with
risk aversion equal to the wealth-weighted harmonic mean of the individual agents’ risk aversion,
K
X Wk
1
=
RAk ,
RAA
WA
k=1
WA =
K
X
Wk .
k=1
Harmonic means are never larger than arithmetic means and are equal only when all individuals
are equally risk averse. Therefore, the risk aversion of the representative investor (of the aggregate
economy) is less than the wealth-weighted average of the individuals’ risk aversion. There are
individual agents with risk aversion even higher than that of the representative agent! If the risk
aversion of the representative agent is already deemed to be high, then there will be individuals with
implausibly high levels or risk aversion. In this framework, the finding a of a high risk premium in
the aggregate leads to an even more implausible conclusion about the individual agent.
In contrast, our analysis shows that for an aggregate economy to display a high degree of risk
aversion and to demand a relatively high risk premium, individuals do not need to have implausibly
high risk aversion.
Prior to concluding, we should make a final technical remark. The results of the previous
two theorems do not require perfect competition within each wealth class. It is sufficient to have
an infinite number of wealth classes consisting of only one agent. The assumption of perfect
competition within each wealth class is made merely to rely on our previously established results.
15
4
Conclusion
The relation between risk preferences of individual agents in the economy and the attitude toward
risk in the aggregate is an important one in financial economics. Ultimately, it is preferences in the
aggregate that determine the relationship between risk and return, and the pricing of risk in the
economy. Yet, it is biases of individuals that characterize deviations from canonical fully rational
financial models. We show that there may be dramatic, significant differences, in the nature of
risk preferences of individual investors and aggregate preferences toward risk in the economy. To
demonstrate this, the paper investigates aggregate properties of an economy where all investors
have convex utility function. First, it is shown that when all investors have identical wealth,
assuming a budget constraint and under perfect competition the aggregate indifference curve is
convex (linear). This result shows that the assumptions of perfect competition and the budget
constraint that eliminates unbounded borrowing or short selling may be of the same degree of
importance as the assumption on the shape of the individual utility function.
We then consider an economy where all agents have convex utility function, but different initial
levels of wealth. If there is a finite number of different wealth classes then the aggregate indifference
curve is continuous, convex, but not differentiable because it consists of finite number of linear
segments. We prove that with a continuum of wealth classes, each being characterized by perfect
competition, the aggregate indifference curve is convex and differentiable. The theorem shows that
an economy that consists of small (atomistic) risk-seeking individual investors in the aggregate
is characterized by an indifference curve consistent with risk aversion. The result shows how
aggregation of individual agents each with a convex utility function can yield an indifference curve
consistent with concave utility function.
We then go a step further. We start with an aggregate convex indifference curve (which corresponds to the case of risk aversion). We show that there exists an economy composed of risk
seeking individuals and a distribution of wealth such that in the aggregate the economy produces
the same risk averse indifference curve as given. This result shows that empirical studies based
on aggregate data can potentially be consistent with a wide variety of individual investor behavior
specifications, even the ones based on utility functions with convex regions. Therefore, caution
must be taken when drawing conclusions about individual behavior based on aggregate data.
We began this paper listing several utility functions that include both risk seeking and risk averse
16
regions (Friedman and Savage 1948, Markowitz 1952, Kahneman and Tversky 1979, Tversky and
Kahneman 1992).
Each of these utility functions were created to describe individual behavior.
What is still unknown, concerning these functions, is how these functions aggregate. Do the risk
seeking regions remain in the aggregate or do they vanish, as they do in this paper? We believe
that this paper is an important stepping stone in exploring the aggregate properties of these other
utility functions.
17
A
Appendix
A.1
Utility Aggregation: Identical Agents
Proof of Proposition 1. Efficient Allocation for Identical Agents. The maximum quantity
of X that any agent can hold is xmax : u(xmax , 0) = k. Notice that each yi = Ψ (xi ; k) is concave
and continuous for all xi ∈ [0, xmax ]. This is so because, by Implicit Function Theorem,
uxx u2y − 2uxy ux uy + uyy u2x
d2 yi
=
−
< 0.
u3y
dx2i
Note that uxy = 0 because we study additively separable utility function. Since the sum of concave
functions is concave, the interior optimum of (1) is a maximum and not a minimum. Due to the
non-negativity constraints, the set of feasible solutions is compact and therefore, the solution exists
and the solution must be located on the boundary of the feasible solution set.
Suppose X = xmax . Then the only corner solution available is associated with xmax being given
to only one agent. Hence, the first agent holds (x1 , y1 ) = (xmax , 0) and the remaining N − 1 agents
hold (xi , yi ) = (0, Ψ (0; k)) for i = 2, ..., N .
Now suppose the economy has X = 2xmax . We already know that it is better to allocate to
the first agent (x1 , y1 ) = (xmax , 0) than to divide xmax amongst several. This leaves us with an
economy with N − 1 agents and only xmax left to distribute. But this is the same as the original
problem (except with one less agent). Hence, the second agent is allocated (x2 , y2 ) = (xmax , 0).
All remaining agents receive (xi , yi ) = (0, Ψ (0; k)) for i = 3, ..., N .
Finally, suppose X = (n − 1)xmax + r is available. By the above argument, the first n − 1 agents
will receive (xmax , 0). The problem is now reduced to allocating r amongst the remaining N −n+1
agents. The corner solution is no longer associated with (xmax , 0) since r ≤ xmax . The corner is
now at the point (x, y) = (r, Ψ (r; k)) and one agent will receive all of r while the remaining agents
receive none of the endowed X.
Therefore, for X = (n − 1)xmax + r, for n = 1, ..., N , n − 1 agents will hold (xi , yi ) = (xmax , 0),
the nth agent will hold (xi , yi ) = (r, Ψ (r; k)),and the remaining N − n − 1 agents will hold (xi , yi ) =
(0, Ψ (0; k)). This is the only corner solution such that all Xmax is allocated and hence, this must
be the optimal allocation. This allocation is efficient by construction.
Proof of Proposition 2: Aggregate Indifference Curve. To model the increasing size of
18
the economy. we start with a fixed integer N ∗ > 0 that represents the number of agents. Agents
in this initial economy achieve level of utility k◦ . The goal is to pass to the limit by increasing
the number of agents while holding Ymax = N ∗ · Ψ (0; k◦ ) fixed. Holding the endowment fixed
while increasing the total number of agents in the economy implies that each agent will be initially
endowed with a decreasing quantity of Y . This causes the initial level of utility of each agent to
decrease. Instead of taking the limit as N grows without bound, we introduce the variable z, z > 0,
and write zN ∗ where z is allowed to grow to infinity. The agents’ decreasing utility is described by
the function k : z → k (z) defined implicitly through the equation
Ymax = zN ∗ · Ψ (0; k (z))
where Ymax = N ∗ · Ψ (0; k◦ ) .
This definition of k (z) assures that as we increase the number of agents (passing to the case of
perfect competition by letting z → +∞), we do not cause the endowment in the economy to
increase without bound; Ymax remains constant for all z. It must be the case that
dk(z)
dz
< 0.
Similarly, in the initial economy xmax = Ψ−1 (0; k◦ ), and in general xmax (z) = Ψ−1 (0; k (z)), and
Xmax = N ∗ Ψ−1 (0; k◦ ). In order to fix the amount of X as we increase the number of agents, we
introduce function f (z) such that as we write zN ∗ , Xmax =
Xmax (z)
f (z)
stays fixed. The function f (z)
counteracts the growth caused by increasing z. Solving for f (z) gives
f (z) =
zΨ−1 (0; k(z))
zN ∗ Ψ−1 (0; k(z))
=
.
N ∗ Ψ−1 (0; k◦ )
Ψ−1 (0; k◦ )
The demand function can now be written as
¸
¶
µ
∙
Xf (z)
−1
; k (z)
Yz (X) = Ψ Xf (z) − Ψ (0; k (z)) int
Ψ−1 (0; k (z))
µ
¸¶
∙
Xf (z)
+ zN ∗ − 1 − int
Ψ (0; k (z))
−1
Ψ (0; k (z))
µ
¸
¶
∙
zΨ−1 (0; k(z))
zX
−1
=Ψ X
− Ψ (0; k (z)) int
; k (z)
Ψ−1 (0; k◦ )
Ψ−1 (0; k◦ )
µ
¸¶
∙
zX
+ zN ∗ − 1 − int
Ψ (0; k (z))
−1
Ψ (0; k◦ )
for X ∈ [0, Xmax ] .
To prove that as z → ∞, Yz → Y∞ uniformly, we must show that for any ² > 0 there exists a
z ∗ > 0 such that for all z ≥ z ∗ , |Yz − YN ∗ | < ². So, for any ² > 0, we will show that such a z ∗
19
exists. Consider the following:
¯ ³
i
´
h
−1
¯
zX
−1 (0; k(z))int
|Yz − Y∞ | = ¯Ψ X zΨΨ−1(0;k(z))
;
k(z)
−
Ψ
−1
(0;k◦ )
)
¯
³
h Ψ (0;k◦i´
¯
◦)
zX
∗
∗
+Ψ (0; k(z)) zN − 1 − int Ψ−1 (0;k◦ ) + ΨΨ(0;k
−1 (0;k ) X − N Ψ (0; k◦ )¯
◦
¯ ³
i
´
h
−1
¯
= ¯Ψ X zΨΨ−1(0;k(z))
; k(z)
− Ψ−1 (0; k(z))int Ψ−1zX
(0;k◦ )
(0;k◦ )
¯
i
h
¯
Ψ(0;k◦ )
+
X
−Ψ (0; k(z)) − Ψ (0; k(z)) int Ψ−1zX
−1
(0;k◦ )
Ψ (0;k◦ ) ¯
The second equality above comes from choosing k(z) so that zN ∗ Ψ (0; k(z)) = N ∗ Ψ (0; k◦ ). Now,
−1 (0;k )
◦
we can decompose z as z = I Ψ
X
−1 (0;k )
◦
+re where I is any non-negative integer and re ∈ [0, Ψ
X
).
Decomposing z in this way, we can now write
¸
∙
µ −1
¶¸
∙
X
Ψ (0; k◦ )
zX
= int
I
+ re
int
Ψ−1 (0; k◦ )
Ψ−1 (0; k◦ )
X◦
∙
¸
X
= int [I] + int
re
−1
Ψ (0; k◦ )
X
.
= I = (z − re ) −1
Ψ (0; k◦ )
Therefore,
|Yz − Y∞ |
¯ ³
´
−1
¯
X
−1 (0; k(z))(z − r )
= ¯Ψ X zΨΨ−1(0;k(z))
−
Ψ
;
k(z)
e
−1
(0;k◦ )
Ψ (0;k◦ )
+
−Ψ (0; k(z)) − Ψ (0; k(z)) (z − re ) Ψ−1X
(0;k◦ )
¯ ³
´
−1
−1
¯
e
= ¯Ψ X zΨΨ−1(0;k(z))
− X zΨΨ−1(0;k(z))
− X Ψ−1r(0;k
; k(z)
(0;k◦ )
(0;k◦ )
◦)
¯
¯
Ψ(0;k◦ )
X
Ψ−1 (0;k◦ ) ¯
¯
¯
Ψ(0;k◦ )
re
X
+
X
+
X
−Ψ (0; k(z)) − zΨ(0;k(z))
−1
−1
−1
Ψ (0;k◦ )
Ψ (0;k◦ )
Ψ (0;k◦ ) ¯
¯ ³
´
¯
re Ψ(0;k(z))
◦)
e
= ¯Ψ X Ψ−1r(0;k
;
k(z)
− Ψ (0; k(z)) − ΨΨ(0;k
−1 (0;k ) X + Ψ−1 (0;k ) X +
◦)
◦
◦
¯ ³
¯
´
¯
re Ψ(0;k(z)) ¯
re
= ¯Ψ X Ψ−1 (0;k◦ ) ; k(z) − Ψ (0; k(z)) + Ψ−1 (0;k◦ ) X ¯
¯ ³
¯
´
∗
¯
re∗ Ψ(0;k(z)) ¯
e
≤ ¯Ψ X Ψ−1r(0;k
;
k(z)
−
Ψ
(0;
k(z))
+
X
¯
−1
Ψ (0;k◦ )
◦)
¯ ³
¯
´
∗
∗
¯
re Ψ(0;k(z)) ∗ ¯
e
≤ ¯Ψ X ∗ Ψ−1r(0;k
;
k(z)
−
Ψ
(0;
k(z))
+
X
¯
Ψ−1 (0;k◦ )
◦)
¯
¯
Ψ(0;k◦ )
X
Ψ−1 (0;k◦ ) ¯
The third equality again uses the fact that zN ∗ Ψ (0; k(z)) = N ∗ Ψ (0; k◦ ). Since re is bounded
³
´
e
;
k(z)
−
and Ψ (xi ; k) is closed and bounded, then there exists an re∗ that maximizes Ψ X◦ Ψ−1r(0;k
◦)
Ψ (0; k(z)) +
re
X.
Ψ−1 (0;k◦ )
The same argument holds for the existence of X ∗ that maximizes the
expression. Hence, the above inequality holds for re∗ . Finally, to finish the proof, we note that as
z → ∞, Ψ (xi ; k(z)) → 0 monotonically. Thus, by the intermediate value theorem there exists a
z ∗ such that for ² > 0
¯ µ
¯
¯Ψ X ∗
¯
¯
¶
re∗
re∗ Ψ (0; k(z ∗ )) ∗ ¯¯
∗
∗
;
k(z
X
)
−
Ψ
(0;
k(z
))
+
¯ = ².
Ψ−1 (0; k◦ )
Ψ−1 (0; k◦ )
20
Hence, for all z > z ∗ and for all X we have
¯ µ
¯
¶
∗
∗ Ψ (0; k(z))
¯
¯
r
r
e
e
∗
∗
¯Ψ X
; k(z) − Ψ (0; k(z)) +
X ¯¯ < ².
¯
−1
−1
Ψ (0; k◦ )
Ψ (0; k◦ )
A.2
Utility Aggregation: Agents with Different Initial Wealth
Proof of Proposition 3: Two Classes of Agents. The proof for the first part is exactly the
same as the proof of Proposition 1 with the exception that we must show that the type 1 agents
are allocated X before the type 2 agents. As before, each type 1 agent can hold a maximum of
x1,max of X and each type 2 agent can hold a maximum x = x2,max of X. Since k1 < k2 , then
x1,max < x2,max . Suppose X = ² where ² > 0 is an arbitrarily small real number. We already
know that the efficient allocation is to give ² to one agent. According to the objective function, we
choose the agent by determining which is willing to give up more of his allocation of Y for ². Said
differently, we give ² to the agent who has the greatest rate of substitution. Thus, since we have
0 (0)| > |y 0 (0)|, the first ² goes to a type 1 agent.
|y1i
2i
0 (0)| > |y 0 (0)| holds for an arbitrary u (x , y ) that satisfies our conditions of
The inequality |y1i
i i
2i
differentiability, monotonicity, and convexity. Since utility function is additively separable, ux (·) is
a function of x only, and uy (·) is a function of y only. Both partial derivatives are positive because
u (xi , yi ) is an increasing function. By the implicit function theorem,
¯
¯
dy2i ¯¯
ux (x)
ux (x)
dy1i ¯¯
and
.
=−
=−
¯
¯
dx x=0,k=k1
uy (y)
dx x=0,k=k2
uy (y)
Since the derivative is evaluated at x = 0, and the agents in the two classes have different utility
with k1 < k2 , it must be that the only difference is in the allocation of y. Then, the nominators are
equal because they are a function of x only. By convexity of u (xi , yi ), the partial derivative uy (y)
is higher for higher values of y. Then, the denominator is larger in the second ratio and therefore
¯
¯
¯
¯
¯
¯ ¯
¯
¯ dy ¯¯
¯ ¯¯ dy1i ¯
¯
¯ 0
¯ ¯ 0
¯
¯ 2i ¯
¯
¯
¯y1i (0)¯ = ¯
>
y
(0)
=
¯
¯
¯
2i
¯ dx ¯x=0,k=k1 ¯
¯ dx ¯x=0,k=k2 ¯
Now, suppose an additional ² is added to the economy. Then, we can give this additional amount
to the type 1 agent who just received ² or to a type 2 agent. Since each agents’ demand function
21
is concave and decreasing, the rate at which each agent is willing to give Y for each additional
increment of X increases,
¯
¯
¯
¯
¯
¯ dy ¯¯
¯ ¯
¯ ¯
¯ ¯¯ dy1i ¯
¯
¯
¯ 0
¯
¯
¯ ¯ 0
1i
0
¯y1i
¯
¯
¯=¯
¯y1i (²)¯ = ¯
>
(0)
¯
¯ > y2i (0)¯ .
¯
¯
¯ dx x=²,k=k1 ¯
¯ dx x=0,k=k1 ¯
The first inequality holds true because ux (x) is increasing by convexity of u (xi , yi ). Therefore, the
X consumption good will be allocated to the type 1 agents until all agents receives x1,max or the
supply of X has been depleted. If all type 1 agents have x1,max , then the remaining supply of X
will be allocated to the type 2 agents one agent at a time with each type 2 agent holding no more
than x2,max .
Proof of Proposition 4: Aggregate Indifference Curve for Two Classes of Agents.
The proof is identical to the proof of Proposition 2. Let X be the amount of the X good available.
If X ≤ N1 x1,max , then the problem is the same as having only the type 1 agents since the type 2
agents will not receive any X. When X > N1 x1,max , all the type 1 agents have x1,max and so the
problem reduces to having only the type 2 agents.
Proof of Theorem 5: Convexity of Aggregate Indifference Curve. We start the proof by
choosing some partition of ki s, for i = 1 to n, over the interval [kmin , kmax ] such that ki < ki+1 . As n
increases, we choose a new partition such that lim |ki−1 − ki | = dki , lim ki = kmin , lim kn = kmax .
n→∞
n→∞
n→∞
Define a continuous and differentiable function f : [0, Xmax ] → [kmin , kmax ] such that f (0) = kmin ,
f (Xmax ) = kmax , and f 0 (X) > 0 with 0 < f 0 (X) < ∞ for all X ∈ [0, Xmax ].
Define ki = f (xi ).
Hence, the partition of ki s is associated with some partition of xi s
on [0, Xmax ] defined by f −1 (ki ) (note that the inverse does exist since f is strictly increasing).
¯
¯
By the continuity of f −1 (k), since lim |ki−1 − ki | = dki , then lim ¯f −1 (ki−1 ) − f −1 (ki−1 )¯ =
n→∞
n→∞
lim |xi−1 − xi | = dxi .
n→∞
For each i, we have
(xi ))
Y (X) = − ΨΨ(0;f
−1 (0;f (x )) [X − xi−1 ] + yi−1
i
for
X ∈ [xi−1 , xi ] .
Evaluating this expression at X = xi and letting n → ∞ we get
(xi ))
dY (xi ) = − ΨΨ(0;f
−1 (0;f (x )) dxi
i
22
for
xi ∈ [0, Xmax ] .
That is, the partition on X becomes infinitely fine and each Yi is defined over an infinitely small
interval. Thus Y becomes differentiable. Further, convexity comes directly from the assumption
Ψ (0; ki+1 )
Ψ (0; ki )
> −1
Ψ−1 (0; ki )
Ψ (0; ki+1 )
for ki < ki+1 . As x ∈ [0, Xmax ] increases, f (x) increase and
dY
dx
approaches zero. Therefore,
dY
dx
=
(x))
− ΨΨ(0;f
−1 (0;f (x)) decreases at a decreasing rate. Hence, the function, Y (x), is convex and differentiable
on the interval [0, Xmax ].
Proof of Theorem 7: Construction of Economy with Risk Seeking Agents. In the
proof we will show how to start with an aggregate convex indifference curve and construct wealth
distribution in an economy where each individual has a concave indifference curve. Define a partition over [0, Xmax ] as xi = i Xmax
n , for i = 0 to n. For each xi , let yi = g(xi ) where g is as defined
in the theorem. We will show that the function g(x) can be reproduced from a system of concave
indifference by carefully choosing a distribution of wealth, ki s. From (3), we can write
i)
Yi (X) = − ΨΨ(0;k
−1 (0;k ) [X − xi−1 ] + yi−1
i
for
X ∈ [xi−1 , xi ] .
We choose ki so that each function Yi (X) is a chord connecting the points (xi−1 , g(xi−1 )) and
(xi , g(xi )). Hence,
−
g(xi ) − g(xi−1 )
Ψ (0; ki )
=
.
Ψ−1 (0; ki )
xi − xi−1
The above statement simply states that choosing ki carefully, the line segment Y (xi ) can be made
to have the same slope as the chord connecting (xi−1 , g(xi−1 )) and (xi , g(xi )).
Such a ki exists
Ψ(0;ki+1 )
and 0 < g 0 (X) < ∞, we can choose a range of ki s
Ψ−1 (0;ki+1 )
i−1 )
for all xi ∈ [0, Xmax ]. That is, we can find
sufficiently large so as to match any slope g(xxi )−g(x
i −xi−1
i)
a ki sufficiently small (large) so as to make the slope ΨΨ(0;k
−1 (0;k ) as steep (shallow) as we need.
i
since from the conditions
Ψ(0;ki )
Ψ−1 (0;ki )
>
Substituting into Yi (X) gives
Yi (X) =
³
g(xi )−g(xi−1 )
xi −xi−1
´
[X − xi−1 ] + g(xi−1 )
for
X ∈ [xi−1 , xi ] .
Yi , for i = 1 to n, represents a point in a sequence of systems of linear equations. Each system of
equations consists of n chords connecting points on g(x).
23
We now need to show that this choice of ki causes the sequence of systems to converge to g(X)
¢
¡
as n → ∞. For any i, choose any x ∈ (xi−1 , xi ). We can rewrite this point as x = xi−1 + r Xmax
n
for r ∈ (0, 1). For all δ > 0, let n > n0 =
|xi−1 − x|
Xmax
δ
¯
¢¯
¡
¯
= ¯xi−1 − xi−1 − r Xmax
n
¢
¡
= r Xmax
n
¡
¢
≤ Xmax
n
³
´
≤ Xnmax
0
=δ
Hence, for all δ > 0, there exists an n0 such that for all n > n0 , |xi−1 − x| < δ. By the continuity of
g(X), for every ² > 0, there exists a δ(n0 ) > 0 such that if |xi−1 − x| < δ, then |g(xi−1 ) − g(x)| < ².
Said differently, for every ² > 0, there exists an n0 such that for all n > n0 , |xi−1 − x| < δ(n0 ), and
hence, |g(xi−1 ) − g(x)| < ². So, let ² > 0 and choose n0 to satisfies the continuity definition. For
all n > n0 ,
|Yi (x) − g(x)|
¯³
¯
´
¯ g(xi )−g(xi−1 )
¯
[x − xi−1 ] + g(xi−1 ) − g(x)¯
=¯
xi −xi−1
¯³
¯
´
¯
¯
i−1 )
= ¯ g(xxi )−g(x
[x
−
x
]
+
(g(x
)
−
g(x))
¯
i−1
i−1
i −xi−1
¯³
¯
´
¯
¯
i−1 )
≤ ¯ g(xxi )−g(x
δ + ²¯
i −xi−1
≤ ².
The last inequality is true since g 0 (X) < 0. But, this is true for any i and for any x ∈ (xi−1 , xi ).
Therefore, for n large enough, each chord, Yi , can be made arbitrarily close to g(X) and hence, the
sequence of systems of linear equations approaches g(X) on [0, Xmax ] uniformly as the sequence
approaches infinity.
24
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27
yi
yi,max
A
B
xA
xB
xi
xi,max
Figure 1: Indifference curve for risk seeking agent.
Y
Ymax
A
Agent 1
Agent 1 holds all X
Agent 2 holds all Y
B
Agent 2
XA
XB
Xmax
Figure 2: Aggregate indifference curve for economy with two
agents.
X
Y
Ymax
X
Xmax
Figure 3: Aggregate indifference curve for economy with five
agents.
y
y2,max
y1,max
Rich
Poor
x1,max
x2,max
Figure 4: Indifference curves for 2 risk seeking agents with
y
y
heterogeneous wealth such that 1,max > 2,max .
x1,max x2,max
x
Y
Poor: NP=2
Rich: NR=2
X
Figure 5: Aggregate indifference curve for economy with
four agents: two poor agents and two rich agents.
Y
Ymax
Xmax
X
Figure 6: Aggregate indifference curve for an economy with
many wealth levels.